jmc

We seek the number $n$ such that $\frac{3+n}{5+n}=\frac{5}{6}$. Multiplying both sides by $5+n$ and by 6 gives $(3+n)(6)=5(5+n)$. Expanding both sides gives $18+6n=25+5n$. Simplifying this equation gives $n=\boxed{7}$.